Discrete Time Markov Processes

A wharf-side fish market sells live fish from its tank during the afternoon. The tank can hold at most 12
fish. During the morning, fish are caught from local waters and used to restock the tank. The number of
fish caught, F, has the following mass function.
0.1 0
0.2 1
0.3 2
( ) 0.2 3
0.1 4
0.1 5
F
f
f
f p f f
f
f
 =
 = 
 = =  = 
 = 
 =
If there is not enough room in the tank for the number of fish caught, then the excess is sold to local
restaurants for $12/fish. Consumer demand during any day (D) for live fish has the following
probability mass function.
0.2 0
0.2 1
( ) 0.3 2
0.2 3
0.1 4
D
d
d
p d d
d
d
 =
 = 
 =  =
 = 
 =
The cost to catch each fish is about $4.00/fish caught plus a fixed fuel surcharge of $20/day. The
selling price out of the tank is $20.00. Assume a Markov model is desired where the state of the
system is defined to be the number of fish in the tank at the end of each day.
a) Create the P matrix for this problem.
b) Solve for the steady state probabilities (πi’s) for your model.
c) Answer the following questions using these πi’s as needed. (Subliminal hint: Don’t forget about
the Law of Total Probability.)
1. What is the expected number of fish in the tank?
2. What is the expected daily cost of getting fish?
3. What is the revenue from fish sold to restaurants?
4. What is the expected number of lost sales per day?
5. What is the expected daily demand?
6. What is the expected daily revenue from sales to customers?
7. What is the expected net revenue per day for this process?

 

 

 

 

 

 

Sample Solution

mentation, systolic BP of 100mHg or less, and respiratory rate >22/min, provides a simple bedside criteria to identify adult patients with suspected infection whose condition are likely to worsen. Additional screening includes a multivariable logistic regression identified that any 2 of 3 clinical variables – GCS 13 or less, systolic BP 100mmHg or less, and respiratory rate 22/min or greater.
Second on the sepsis continuum, the patient needs to present 2 of the SIRS criteria and a positive culture of sputum, blood, or urine that reflects growth of bacteria.
Severe sepsis is classified when there is sepsis and hypo-perfusion with organ dysfunction that is unresponsive to fluid resuscitation. It has more recently been viewed as endothelial dysfunction resulting from overwhelming inflammatory mediation, in conjunction with profound, unopposed coagulation. The capillary vasculature sustains a significant injury due to the cascade of events that ends in capillary occlusion. The greater the occlusion, the greater risk for organ failure because cellular level circulation requires a functional capillary network for delivery of oxygen and nutrients and removal of cellular metabolic waste products.
When infection or injury prompts an initially widespread inflammatory response (SIRS), the normally smooth surface of the microvascular endothelium is roughened and damaged by the response. In addition, the release of inflammatory mediations promotes vasodilation with an increase in capillary permeability. This causes little holes in the endothelium that the systemic mediators try to facilitate the healing of. The four main factors associated with severe sepsis is hyperinflammation, hypercoagulation, microvascular obstruction, and endothelial responses that leads to accelerated formation of microclots on the non-smooth surface of the endothelium. This consumes plates and inhibits clot lysis. This p

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